Simple Proof Help

[cubixguy77]

Prove that if a ε F, v ε V, and av = 0, then a = 0 or v = 0.

Seems simple and obvious but i can't put it into words. Thanks for the help.

[Identity]

If a does not equal zero and v does not equal zero then av equals some number x which is not equal to zero.
Therefore, for av = 0
either a or v equals zero or both a and v equal zero.

[HallsofIvy]

Prove that if a ε F, v ε V, and av = 0, then a = 0 or v = 0.

Seems simple and obvious but i can't put it into words. Thanks for the help.

What are F and V? Not only the "proof" but even whether that is true depends on what you are talking about there! Is V a vector space and F the underlying field? In that case: any member of a field, other than 0, has a multiplicative inverse. If a is not 0, then we can multiply both sides of the equation by 1/a and have (1/a)(av)= (1/a)(a) v= v= (1/a)v= 0. That is, if a is not 0, v must be 0: if av= 0 then either a or v or both must be 0.